Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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If two invertible matrices A and B commute, then A^-1 and B^-1 must commute as well ??

If two invertible matrices A and B commute, so their inverse must commute as well or not ?
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40 views

How to calculate inverse of Variance Gamma call price formula using Newton-Raphson search

The Variance Gamma call price formula is given by: $$C(0)= \int\gamma(R) e^{-rT} \int f\left(S(0) e^{\theta R+\omega T+\frac12 \sigma^2 R} e^{rT-\frac12 \sigma^2 R+\sqrt{T}\sqrt{R/T} \sigma ...
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25 views

Derivative of inverse cosecant?

I am slightly confused by this, because when I worked out the derivative of arccosec(x), my answer was $\frac{-1}{x\sqrt{x^2-1}}$, which agrees with the answers online. However this would imply that ...
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293 views

Inverse of the sum of the inverse of two matrices

I need to compute $ (A^{-1} + B^{-1})^{-1} $. Both $A$ and $B$ are symmetric and $A$ is invertible and PSD. I already know $B^{-1}$ and $A$, but I don't have $A^{-1}$ and $B$. Is there a formula to ...
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54 views

Relation between two inverses

Suppose you know $(I+T)^{-1}$, is there any way for approximate the inverse of the matrix $(I+\alpha T)^{-1}$, where $\alpha\in{\mathbb{R}}$?
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Inverse of matrix with 1 in diagonal and some entries above them.

Suppose matrix N has a,b,c above the main diagonal, and all other entries equal to $0$. that is, $N=\begin{bmatrix} 0 & a & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & c ...
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23 views

Inverse function for a surface of revolution

I have the following function: $$ f(x)= c_1\cdot c_2\cdot x\cdot \arctan\left(c_2\cdot x\right)-\frac{1}{2}\cdot c_1\cdot \ln\left(1+c_2^2\cdot x^2\right) $$ with $c_1=0.003$ and $c_2=150$ constants ...
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77 views

Finding only first row in a matrix inverse

Let's say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what's the best algorithm to use to calculate just this row? ...
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34 views

Evaluate cos[(1/2)[arcsin(-3/5)]]. I'm not sure what i'm doing wrong.

$x=\arcsin(-3/5), \; \sin x = -3/5$ **Drew a triangle to find $\cos x$ $\cos x = 4/5$ Now, I don't know what to do from here. I know I have to use a double angle formula, but when I evaluate the ...
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0answers
51 views

a special matrix inverse

Let $A=\left( \begin{matrix} {{A}_{11}} & \ldots & {{A}_{1n}} \\ \vdots & \ddots & \vdots \\ {{A}_{n1}} & \cdots & {{A}_{nn}} \\ \end{matrix} \right)$ be an ...
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23 views

Inverse of function with two Exponential Eulers Terms

How can I go about getting the inverse of$ f(t) = e^{-.001t}\cdot e^{-.005t}$? I have found a couple of calculators online that suggest that the answer is: $t=-166.667\ln(y)$, but I would like to know ...
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2answers
51 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function ...
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29 views

How to solve an inverse relationship (cooking temp/time)

How to figure out exactly the "add a little more time" to the question: cook at 425 deg for 18 minutes ... if I have several things in the same oven and need to set the oven at 375. I can't use a ...
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48 views

Matrix derivatives of determinant and inverse related to $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$

I would like to calculate the derivatives of determinant and inverse related to the term $\mathbf{X}\mathbf{X}^{T}+\mathbf{C}$ with respect to $\mathbf{X}$, where $\mathbf{C}$ is a constant matrix. ...
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36 views

help with inverse function in $\mathbb R^2$

$F(x,y)=(x^2+2y^2,2x^2+y^2)$, and $A=\{(x,y):x>0,y>0\}$ I need to show $F(A)=\{(u,v):0<u/2<v<2u\}$ I also need to find what is $G(=F^{-1}):B\rightarrow A$ For the first question I ...
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19 views

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$

$F(x,y) = (x^2 + 2y^2, 2x^2 +y^2)$ and $A= \{(x,y) : x>0, y>0\}$ I need to find the following: $(a)$ Show $F$ is one-to-one on $A$. $(b)$ Show that $F(A) = \{(u,v) : 0 < \frac{u}{2} < v ...
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1answer
31 views

Inverse symmetric circulant matrix

I want to inverse a very particular matrix numerically. The matrix is always symmetric and circulant. As an example of a 4x4 matrix I would want to inverse \begin{pmatrix} v_0 & v_1 & v_2 ...
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1answer
34 views

Inverse Function Theorem when determinant is undefined

For $f(x,y) = (x^3 - y^2, \sin{x} - \ln{y})$ f-inverse exists and is differentiable in a non-empty set around $(-1,0)$. Find $D(f^{-1})$ at $(-1,0)$. Seemingly this is an Inverse Function Theorem ...
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29 views

Inverse Image Proof

Let $f:X\rightarrow Y$. Let $A$, $A_1$ and $A_2$ be subsets of $X$ and $B$, $B_1$, and $B_2$ be subsets of $Y$. Then, I need to prove that $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$. I know ...
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34 views

What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$?

Let $f:\mathbb{R}^3 \to \mathbb{R}^3$ be $$f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right).$$ My question is what restrictions are necessary on ...
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54 views

Inverse of matrix mod $26$ wolframalpha wrong

I want to find $A^{-1} \pmod{26}$ for $A=\begin{bmatrix}10&3\\5&3\end{bmatrix}$ and I did the conventional $\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ and found the ...
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26 views

Is there any way to test the existence of left or right inverse matrix?

I know that the inverse matrix of a square matrix exists iff its determinant isn't 0. What about a non-square matrix? Is there any theorem about the existence of a ...
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79 views

Prove $\frac1{\sqrt x}$ is continous on $(0,\infty)$. Stuck on last line!

Let $f(x) = \frac1{\sqrt x}$ for $x\in(0,\infty)$. Given $\varepsilon>0$ and $x_0\in(0,\infty)$, show there exists $\delta>0$ such that $$|x-x_0|<\delta$$ implies that $$|f(x)-f(x_0)| ...
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33 views

Finding the domain of a difficult inverse

$f(x)=\frac{3x+5}{-6x+2}$ , largest possible domain Find $f^{-1}(x)$ of this 1-1 function and the domain. So I wrote the equation as $$y=\frac{3x+5}{-6x+2}$$ Interchanged x and y, and made y ...
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322 views

how to find two right-inverse functions of a function

i am stuck in this problem. i need to find two right-inverse functions of this function: $h: \Bbb N_0\times \Bbb N \to \Bbb N, (m,n)\mapsto m+n$. i know that the function h' is a right inverse of ...
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200 views

Left inverse of a function

Let $f$ be the function $f\colon \mathbb{N}\rightarrow\mathbb{N}$, defined by rule $f(n)=n^2$. Needed to find two left inverse functions for $f$. I know only one: it's $g(n)=\sqrt{n}$. Does anyone ...
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44 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
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38 views

Application of Inverse Function Theorem

This is a seemingly easy exercise. Yet I am not sure if I am missing any finer details here as this is listed as one of the challenging problems on Dr. Epstein's (Upenn) course site for real analysis. ...
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Is $T(M)=PMP^{-1}$, where $P=\begin{bmatrix}2&3\\5&7\end{bmatrix}$ linear? If so, how to prove?

If I define $\vec{v}=\begin{bmatrix}a\\b\end{bmatrix}\text{and }\vec{w}=\begin{bmatrix}c\\d\end{bmatrix}$, I end up getting ...
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22 views

Row sums of inverse of PageRank matrix variant

In the book "Deeper inside Pagerank" (Amy N. Langville and Carl D. Meyer), (http://www.ulco.nl/docs/Langville.pdf), in the page 352 of the book (page 18 of the document in url), it is stated that "The ...
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1answer
47 views

What does it mean for a matrix to change basis?

I understand what it means for vectors, i.e. $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} ...
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1answer
35 views

$y=N(N^TT N)^{-1}N^TT$

Let $T$ be a square $n\times n$ matrix. This matrix is symmetric and positive definite. Let $N$ be a $n\times s$ matrix where $s<n$. I want to be able to compute: $$y=N(N^TT N)^{-1}N^TT$$ I can ...
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1answer
30 views

How to calculate frequency with clock signal is 500ps in digital logic?

How can i calculate frequency if clock signal 500ps. I know the only formula, that is T=1/f But i cant able to calculate, can ...
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137 views

Understanding inverse of a function

I was trying to understand the proof for the following proposition. Proposition: If $\{f_n\}$ is a sequence of $\bar{\mathbb{R}}$ valued measurable functions on $(X,\mathcal{M})$, then the functions ...
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1answer
71 views

find an inverse function of complicated one

Let $f:\mathbb{R}\rightarrow \mathbb{R}$: $$f(x) = \sin (\sin (x)) +2x$$ How to calculate the inverse of this function? So far i searched a lot in the internet but i didn't find any easy algorithm ...
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1answer
38 views

find the inverse of $\frac{1-e^t}{1+e^t}$

Hi I am trying to prove that the inverse of $f(t) = \frac{1-e^t}{1+e^t}$ is $F^{-1}(t) = \ln\left(\frac{1-t}{1+t} \right )$ But I don't quite know where to start? Do I just sub ...
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73 views

Proof Regarding Determinants of a Matrix

Prove the following statement: If $A$ is an $n$ by $n$ matrix, such that $\sum_{j = 1}^n a_{ij} = 0$, for all $1 ≤ i ≤ n$, then $\det A = 0$ too. (Sorry I don't know how to format this equation) ...
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39 views

Invertible matrix problem

Given three $n \times n$ matrices $A$, $B$ and $C$. Prove that if $AB+AC$ is an invertible matrix then $A$ is also an invertible matrix. How can this be possible? I found that $B=A^{-1}-C$ and when I ...
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1answer
37 views

Inverse Laplace tranform via the table formulas

In my inverse Laplace table there is this inversion "formula": $(1) \frac{1}{s-a} \rightarrow e^{at}$ I understand that $\mathcal{L}^{-1}[\frac{1}{s+4}] = \frac{1}{2}\sin(2t)$ But why can I not do ...
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1answer
55 views

Tan inverse summation

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$ How to simplify this ? I think I will have to use this : but it looks too long a method . Is there a method or symmetrical way which yields ...
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31 views

Sherman–Morrison–Woodbury formula and hollow matrix

Suppose there are two matrices: $A_{n\times n}= \begin{bmatrix} a_0 & 0 &a_1 & \dots \\ 0 & a_1 & 0 &\dots \\ a_1 & 0 &a_2 & \dots \\ \vdots & \vdots & ...
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1answer
786 views

inverse of a covariance matrix 3x3

I have 2 pixels with size 1x3 called $A$ and $B$ and I have to compute the following equation: $$ A^T *(\Sigma+ I_3*\lambda)^{-1}*B $$ where $\Sigma$ is the covariance matrix (3x3) between vectors ...
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1answer
27 views

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x?

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x? Solution $\tan^{-1}x=\tan^{-1}a-\tan^{-1}b=\tan^{-1}\frac{a-b}{1+ab}$ ...
3
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1answer
42 views

Arithmetic modulo primes task

I'm dealing with a problem here. The problem is as follows: There is a set $Z_p=\{0,1,2,3,...,p-1\}$ where $p$ is a prime. From this set we form a new set $B=\{x+x^{-1}\mid x\in Z_p\}$, where the ...
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0answers
83 views

Closed form for elements of inverse matrix of lower triangular matrix of any size

If we have a lower triangular matrix $$A=\left(\begin{array}{rrrrr}a_{1,1}&0&0&\cdots&0\\a_{2,1}&a_{2,2}&0&\cdots&0\\a_{3,1} &1_{3,2}&a_{3,3}&\cdots&0\\ ...
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1answer
20 views

a matrix inverse problem

Given a matrix $X$, let $D$ be a diagonal matrix whose diagonal elements are row sums of $X$, let $I$ be an identity matrix. Now I have a resultant matrix of $Y=(I-X)^{-1}$, and I would like to ...
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1answer
25 views

Deriving an identity using the Woodbury matrix identity

I am working through an algorithm derivation in Kernel Adaptive Filtering: A Comprehensive Introduction by Liu, Principe and Haykin. The part I'm having trouble with is on page 104 if you have the ...
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1answer
13 views

Inversion of Boolean function Application

Asume you have a boolean function $f$ which takes $n$ parameters and gives $m$ results. In addition, you have a boolean function $g$ takes $p$ parameters and gives out $n$ results. You could ...
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48 views

Show that the inverse function to $f(x)=\int_{1}^{x}\frac{dt}{t}$ is differentiable

Show that the inverse function to $$f(x)=\int_{1}^{x}\frac{dt}{t}$$ is differentiable. I know that the integral is $\ln(x)$, but I don't know how to show that it is differentiable in a good way ...
4
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3answers
146 views

Calculate inverse of arbitrarily sized, lower triangle matrix with a specific pattern.

I have a matrix of the following form: $$A=\begin{bmatrix} 2 & 0 & 0 & 0 \\-1 & 2 & 0 & 0\\ 0 & -1 & 2 & 0\\ 0 & 0 & -1 & 2 \end{bmatrix}$$ which, in ...